Cf: Sign Relations, Triadic Relations, Relation Theory • Discussion 5 http://inquiryintoinquiry.com/2021/01/15/sign-relations-triadic-relations-relation-theory-discussion-5/
Re: Cybernetics ( https://groups.google.com/g/cybcom/c/Wyz1oRmturc ) ::: Cliff Joslyn ( https://groups.google.com/g/cybcom/c/Wyz1oRmturc/m/nRfUd8SLBwAJ ) Dear Cliff, I’m still collecting my wits from the mind-numbing events of the past two weeks so I’ll copy your last remarks here and work through them step by step. CJ: <QUOTE> I think what you have is sound, and can be described in a number of ways. In years past in seeking ways to both qualify and quantify variety in systems I characterized this distinction as between “dimensional variety” and “cardinal variety”. Thankfully, this seems straightforward from a mathematical perspective, namely in a standard relational system S = ×_{i=1}^k X_i, where the X_i are dimensions (something that can vary), typically cast as sets, so that × here is Cartesian product. </QUOTE> Relational systems are just the context we need. It is usual to begin at a moderate level of generality by considering a space X of the following form. X = ×_{i=1}^k X_i = X_1 × X_2 × ... × X_{k-1} × X_k. (I’ll use X instead of S here because I want to save the letter “S” for sign domains when we come to the special case of sign relational systems.) We can now define a “relation” L as a subset of a cartesian product. L ⊆ X_1 × X_2 × ... × X_{k-1} × X_k. There are two common ways of understanding the subset symbol “⊆” in this context. Using language from computer science I’ll call them the “weak typing” and “strong typing” interpretations. • Under “weak typing” conventions L is just a set which happens to be a subset of the cartesian product X_1 × X_2 × ... × X_{k-1} × X_k but which could just as easily be cast as a subset of any other qualified superset. The mention of a particular cartesian product is accessory but not necessary to the definition of the relation itself. • Under “strong typing” conventions the cartesian product X_1 × X_2 × ... × X_{k-1} × X_k in the type-casting L ⊆ X_1 × X_2 × ... × X_{k-1} × X_k is an essential part of the definition of L. Employing a conventional mathematical idiom, a k-adic relation over the nonempty sets X_1, X_2, ..., X_{k-1}, X_k is defined as a (k+1)-tuple (L, X_1, X_2, ..., X_{k-1}, X_k) where L is a subset of X_1 × X_2 × ... × X_{k-1} × X_k. We have at this point opened two fronts of interest in cybernetics, namely, the generation of variety and the recognition of constraint. There’s more detail on this brand of relation theory in the resource article linked below. I’ll be taking the strong typing approach to relations from this point on, largely because it comports more naturally with category theory & by which virtue it enjoys more immediate applications to systems and their transformations. But my eye-brain system is going fuzzy on me now, so I’ll break here and continue later ... Regards, Jon Resources ========= • Relation Theory ( https://oeis.org/wiki/Relation_theory ) • Triadic Relations ( https://oeis.org/wiki/Triadic_relation ) • Sign Relations ( https://oeis.org/wiki/Sign_relation ) On 12/31/2020 1:15 PM, Jon Awbrey wrote:
Cf: Sign Relations, Triadic Relations, Relation Theory • Discussion 4 http://inquiryintoinquiry.com/2020/12/31/sign-relations-triadic-relations-relation-theory-discussion-4/ Re: Previous Post https://inquiryintoinquiry.com/2020/12/29/sign-relations-triadic-relations-relation-theory-discussion-3/ Re: Cybernetics https://groups.google.com/g/cybcom/c/Wyz1oRmturc ::: Cliff Joslyn https://groups.google.com/g/cybcom/c/Wyz1oRmturc/m/nRfUd8SLBwAJ Dear Cliff, Many thanks for your thoughtful reply. I copied a transcript to my blog (see link above) to take up first thing next year. Here's hoping we all have a better one! Regards, Jon Cliff Joslyn wrote: <QUOTE> I think what you have is sound, and can be described in a number of ways. In years past in seeking ways to both qualify and quantify variety in systems I characterized this distinction as between “dimensional variety” and “cardinal variety”. Thankfully, this seems straightforward from a mathematical perspective, namely in a standard relational system S = ×_{i=1}^k X_i, where the X_i are dimensions (something that can vary), typically cast as sets, so that × here is Cartesian product. Here k is the dimensional variety (number of dimensions, k-adicity), while n_i = |X_i| is the cardinal variety (cardinality of dimension i, n_i-tomicity (n_i-tonicity, actually?)). One might think of the two most classic examples: • Multiadic diatom/nic: Maximal (finite) dimensionality, minimal non-trivial cardinality: The bit string (b_1, b_2, …, b_k) where there are k Boolean dimensions X_i = {0, 1}. One can imagine k → ∞, an infinite bit string, even moreso. • Diadic infini-omic: Minimal non-trivial dimensionality, maximal cardinality: The Cartesian plane ℝ², where there are 2 real dimensions. There's another quantity you didn't mention, which is the overall “variety” or size of the system, so ∏_{i=1}^k n_i, which is itself a well-formed expression (only) if there are a finite number of finite dimensions. </QUOTE>
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